The following table shows the percentage of individuals in each age group who use an online tax program to prepare their federal income tax return.
AGE ONLINE TAX PROGRAM (%)
18-34 16
35-44 12
45-54 10
55-64 8
65+ 2
Suppose a follow-up study consisting of personal interviews is to be conducted to determine the most important factors in selecting a method for filing taxes.
a) How many 18-34-year olds must be sampled to find an expected number of at least 25 who use an online tax program to prepare their federal financial income tax return?
b) How many 35-44-year olds must be sampled to find an expected number of at least 25 who use an online tax program to prepare their federal income tax return?
c) How many 65+-year olds must be sampled to find an expected number of at least 25 who use an online tax program to prepare their federal income tax return?
d) If the number of 18-34-year olds sampled is equal to the value identified in part (a), what is the standard deviation of the percentage who use an online tax program?
e) If the number of 35-44-year olds sampled is equal to the value identified in part (b), what is the standard deviation of the percentage who use an online tax program?
I'm not exactly sure where to start with answering the question, and my text doesn't have a similar question. What do you think my first step should be? Plus shouldn't the numbers add up to 100%. because they don't.
Thanks
To answer these questions, we need to make certain assumptions and use statistical calculations. The first step is to assume that the given percentages represent the population proportions accurately. Although the percentages do not add up to 100%, we can assume that the missing percentage represents individuals who do not use an online tax program and use an alternative method for filing their taxes.
Now let's address each part of the question:
a) To find the number of 18-34-year-olds who must be sampled to find an expected number of at least 25 who use an online tax program, we need to calculate the sample size using the formula for sample proportion:
Sample size = (Z^2 * p * (1-p)) / E^2
where:
- Z is the critical value for a desired level of confidence
- p is the estimated population proportion
- E is the desired margin of error (in this case, set to 0 because we want at least 25)
In this case, the estimated population proportion for 18-34-year-olds is 16%. We can convert it to a decimal by dividing it by 100: p = 0.16. Assuming Z = 1.96 for a 95% confidence level, the formula becomes:
Sample size = (1.96^2 * 0.16 * (1-0.16)) / 0^2
= (3.8416 * 0.1344) / 0
= undefined (because of division by zero)
However, since we want to find at least 25 individuals, we can increase the sample size until we reach that minimum threshold. Let's try increasing the sample size to 100:
Sample size = (1.96^2 * 0.16 * (1-0.16)) / (25/100)^2
= (3.8416 * 0.1344) / 0.0625
≈ 8.26
Hence, we would need to sample at least 9 individuals in the 18-34 age group to expect finding at least 25 who use an online tax program.
b) The process for finding the sample size for the 35-44-year-old group is the same as in part (a). Using the same formula as above, we substitute p = 0.12 and repeat the steps to find the sample size. Assuming Z = 1.96 again, we can calculate:
Sample size = (1.96^2 * 0.12 * (1-0.12)) / (25/100)^2
= (3.8416 * 0.1056) / 0.0625
≈ 6.48
So, we would need to sample at least 7 individuals in the 35-44 age group to expect finding at least 25 who use an online tax program.
c) Similarly, for the 65+ age group, we can use the formula with p = 0.02 and repeat the steps:
Sample size = (1.96^2 * 0.02 * (1-0.02)) / (25/100)^2
= (3.8416 * 0.0196) / 0.0625
≈ 1.2
Here, we would need to sample at least 2 individuals in the 65+ age group to expect finding at least 25 who use an online tax program.
d) To calculate the standard deviation of the percentage who use an online tax program among 18-34-year-olds, we can use the formula for the sampling standard deviation:
Standard deviation = sqrt((p * (1-p)) / n)
where n is the sample size and p is the population proportion (0.16). The sample size is 9 (calculated in part (a)). Substituting these values into the formula:
Standard deviation = sqrt((0.16 * (1-0.16)) / 9)
= sqrt(0.1344 / 9)
≈ 0.124
Therefore, the standard deviation would be approximately 0.124 for the percentage of 18-34-year-olds who use an online tax program.
e) Similarly, for the 35-44 age group, we can use the same formula as in part (d) but with the sample size calculated in part (b) (7) and population proportion of 0.12. Substituting the values:
Standard deviation = sqrt((0.12 * (1-0.12)) / 7)
≈ 0.115
Thus, the standard deviation would be approximately 0.115 for the percentage of 35-44-year-olds who use an online tax program.
Remember that these calculations are based on the assumptions made earlier, and the sample sizes provided here are only estimates. Actual sampling and statistical analysis should take into account the desired level of confidence and other factors.